As of Today 5

February 14, 2013. We continued with Saxon Math level 4th,  lesson 131.  We have only 10 more lessons to go.  It went much better than I had anticipated.  Maybe because some of the topics we had already encountered  through different curricula, so they were not exactly new. Math operations on decimals and common fractions still needed some polishing, specially when they required changing from improper fraction to mixed one or vice versa .  The need to use a few skills in a sequence confuses Robert.  Just changing fractions from one kind to another was not an issue, but doing this as a next step AFTER  adding fractions baffled him.  Robert also had to be reminded to change  cents to dollars before adding money. He knew that he should place a decimal point under a decimal point, but he did not change cents into decimal part of the dollar.  He used to do that in the past, but since nobody was practicing that with him for at least 6 months, he forgot.

When I look at the lesson for tomorrow, I noticed, than one of the problem required counting circumference of the circle.  That gave me a pause. Many of the previous lessons in Saxon Math dealt with a circle.  Robert drew circles using compass.  He also  drew diameters and radii, and measured their lengths.  Finding circumference calls for an introduction of the number Pi.  Long ago, very long ago, I had been introducing Pi to 38 typical fifth graders.  With the help of the strings we measured the circumferences of the random round objects and divided them by lengths of their diameters.  We all came to some approximation of Pi.

Robert can only divide by numbers up to 12.  So dividing by lengths of the diameters which might be decimals or multiple digit numbers is not an option.

In a few weeks, I will resume teaching geometry through Firelight textbook/workbook and then we will spend more time on counting circumferences and maybe even areas of the circles.  For now, I will stick with a mechanical formula:  Circumference = 3.14 times diameter.

Saxon Math, grade 4th is not an easy curriculum to follow.  Its approach to teaching/learning is to address simultaneously different topics through small steps and increase their complexity (difficulties)  continuously from one lesson to the next.  Most of the teachers and parents do not like such methodology.  It seems that there is not enough opportunities to practice new skills to reach fluency. To address that I kept making my own pages to help Robert practice new kinds of problems.  Before each lesson, I prepare worksheets that  address one or two new skills and give Robert an opportunity to practice them before he confronts them on the page of Saxon Math workbook.

There are, however, big advantages to this approach.  Not once, when I taught typical students and when I taught Robert,  I was confronted by the fact that children who seemed to master one skill through repetitive practice, forgot it as soon as they moved to a different chapter of a textbook/workbook.

For Robert, switching from one kind of problems to another needs to be practiced daily to help with flexibility of thinking and ability to apply his skills in slightly new circumstances.   It is a different thing to practice ten times in a row finding the length of a radius as a half of the length of a diameter than to, for instance, be required to draw a circle knowing its  diameter AFTER  solving a few unrelated arithmetic problems, counting total price, or reading  a graph. While completing a page of similar problems helps with fluency, and thus  cannot be avoided, such completion, even errorless,  cannot be considered a criterion  for mastering the skill.  It is the  application of the skill in a new context, new situation, and without the support of similar examples that attests to a solid learning.

I use the fact that each unit  in Saxon Math (Grade 1 to 4) has two parts/pages: A and B.  Problems in part B match problems from part A requiring the student to use similar technique.  For a student who, like Robert, is very reluctant to independently solve problems that don’t rely on mechanical skills (algorithms) , having an opportunity to look at the problem #3 on page A to solve problem #3 on page B, is a very important step toward controlling his own learning processes.   I don’t mind that Robert looks for clues.  I not only encourage but strongly suggest to go back and read the previous solution and then use similar tactic.  Looking back is an important tool to have as it  allows Robert to loosen his dependence on clues coming from me or his teachers. This is not an independence yet, but it is a half a step toward it.

Twist. When the Teaching Backfire.

This evening Robert was finding distances along the routes made from two or three connected trails.  He did not have any problems with finding distances by adding lengths of two or three segments dividing the entire route.   When, however, he had to tell how many meters one has to make by going from A to B and back, he was lost. I made a drawing. I explained, “This is 700 meters from A to B and this is 700 meters from B to A.  Robert said something, but I did not understand.  So I continued.  “This way is 700m and the way back is 700m.How much together?” I was sure that after hearing the magic word “together”, Robert would immediately come with addition.

Instead I saw complete confusion in Robert’s eyes.  Then I heard separate words: “negative”, “zero”.

At first I did not get it.   Since it was not what I expected (adding or multiplying by 2) I dismissed Robert’s answer without giving it a second thought. Besides, I was preoccupied with finding a better way to explain the problem.  As I contemplated using a string to demonstrate how to measure the length of the round trip, it suddenly dawned on me.

Robert was using his knowledge of positive and negative numbers as I presented them to him on a number line.  He treated both routes as opposite vectors, which had a sum of zero.  Coming back to the starting point meant there was a zero shift. Just like adding 5 + (-5) =0

Robert applied his knowledge of integers in a new context.

I understood the genesis of Robert’s error.  Although Robert did not understand the  basic concept of length he was, nonetheless,  grasping with and applying abstract concepts of positive and negative numbers as they relate to direction on the number line.

There is also the possibility that my emphasis on teaching Robert to add integers led to over generalization of the new rules.

Was that an example of teaching that backfired?  Did Robert lost trust in his own analysis of the problem?

As long as I remember,  Robert has never solved the problem of length of the round trip independently.  So, maybe that was not a fault of my teaching.  Maybe this time, he felt he had a tool to solve the problem and used it, although incorrectly.

I am not sure what to do next.  Should I continue with integers on the number line and introduce the concept of absolute value as a distance from zero, or stick to the string?

I do not know what to do about that.  Should I introduce absolute value or just stick to the string?

Teaching Without Curriculum.

Before I began writing about different math curricula I used  with my son, I have to make a full disclosure about my emotional state in regard to that topic. I feel a lot of anger and confusion.  This anger is the result of those experiences:

1. None of the math textbooks or workbooks I bought for my son was recommended  or even known to school.

2. None of the program my son attended proposed any math curriculum for him. I am grateful that the two schools accepted Saxon Math  which I had been using long before suggesting this program to school.

3.I learned about many math programs from parents’ e-mail lists or later from catalogs, which somehow found a way to my home.

4.The schools tend to use goals (too narrow and too few) written in the IEP  as a reason NOT TO WORK ON ANYTHING ELSE. For a student who has an access to the general education classes, that might be not as confining and disastrous as it is for a student whose whole education has been reduced to IEP goals.

5.Working only on specific math goals in a vacuum, without connecting them to related topics, leads to  many holes in understanding of  the concepts.  Consequently, we demand that children subjected to such approach jump from a stone to a stone while crossing a brook, instead of walking over the bridge made of well-connected and supported boards as their typical peers do with the help of a well designed, comprehensive curriculum.

6.Lack of curriculum forces teachers to search for appropriate pages on internet.  The pages from internet allow for extra practice but not for an introduction of  novel concepts.  Relying on such pages negatively affects teaching, as it leads to mechanical applying of formulas without understanding concepts behind them.  It is not good for the teacher and it is certainly not good for the student. 

7. I suspect, however, that this approach is wholeheartedly supported by the school administration as it saves money.  Instead of buying expensive textbooks and workbooks for students,  the administration relies on teachers to print pages from internet to address narrowly  formulated IEP’s goals. Those goals, I have to emphasize again, are too narrow to result in any meaningful learning. And thus the students who need more to learn, get much, much less than their typical peers.

Not a Laughing Matter

The events, I recorded relatively accurately in Surviving the Doomsday, are serious enough to discuss them further.

1. I said, “Relatively”, because I omitted a few details.  For instance, Amanda made a short comic strip with pictures related to the disappearance of the wallet.  I hoped that seeing someone else taking the wallet would let Robert understand what had happened.  I believed that any explanation would be better than none.

2. It is interesting that having a strong daily routine (studying together with the help of  already prepared worksheets)  helped, at least  temporarily, to deal with the break of another routine/or routine attachment.

3.The fact that Robert accepted so easily  dad’s departure for work seemed to indicate, that it was a calming and reassuring for Robert to know that other routines (and other people routines) remained unbroken.

4. For almost a year before the wallet was gone, I had been concerned about Robert’s attachment to the wallet.  A few times I suggested to him that he should get a new wallet. Robert reacted with a forceful indignation.  So I delayed the time of unavoidable confrontation until the time I would feel calm and strong enough to face it.  I postponed for too long.

5. Had I replaced Robert’s wallet  sooner, a few things might be different:

a.I would be emotionally prepared for the outcome. When the wallet disappeared, I was in a state of panic. That is not a good state when you have to handle unpredictable consequences. I was stressed and it showed.

b. The wallet would not disappear, but be replaced by another one with Robert’s  full knowledge.  Although Robert would still protest vehemently, he would at least know where the old wallet was.  Consequently, Robert’s anxiety would be lower, although his resolve not to give up might be even stronger. I would reduce Robert’s anxiety even at the cost of stronger protests.

c. Having his cards transferred from one wallet to another would make it easier to understand the fact that the new wallet is assigned the same function the yellow wallet had.  Robert would, probably, remove the cards a few times, but the idea that the cards should be in a new wallet would slowly sink in. Placing entirely new card (McDonald’s gift card instead of a bank card and an  ID card which were gone with the yellow wallet) was more like a symbolic gesture than a real transfer.

d. I would give Robert an option of either attending a preferable activity (skiing, eating in favorite restaurant) WITH a NEW WALLET or staying at home.  Given my prior experiences, including the one which I described in Negotiations , it would take a lot of convincing but no more than two hours of time.  After going outside even once with a new wallet, Robert would not have problems taking it again.

e.  Knowing, from experience, that one wallet can be replaced by another, would make Robert’s reaction   to its sudden disappearance weaker and more flexible.

The yellow wallet was an eye sore.  It was also very uncomfortable.  It was difficult to squeeze the cards in or take them out; the money kept falling out.  I should have the courage to convince (?) Robert to replace it sooner. It would not be easy, but it would be much less stressful for all of us than dealing with an unplanned crisis.

The good rule is to intervene as soon as too strong, unhealthy habit is forming.

It is a very good rule indeed. However,  not an easy one to follow.